Time: 9:30 - 11:30,  January 27, 2021

Venue: Room 612, A6, Institute of Mathematics, VAST

Abstract: We classify singularities $fin K[[x_1,ldots, x_n]]$, which are simple, i.e. have no moduli, w.r.t. right equivalence, where $K$ is an algebraically closed field of characteristic $p>0$. For $K=mathbb R$ or $mathbb C$ this classification was initiated by Arnol'd, resulting in the famous ADE-series. The classification w.r.t. contact equivalence for $p>0$ was done by Greuel and Kroening with a result similar to Arnol'd's. It is surprising that w.r.t. right equivalence and any given $p>0$ we have only finitely many simple singularities, i.e. there are only finitely many $k$ such that $A_k$ and $D_k$ are right simple, all the others have moduli. We conjecture a similar finiteness result for singularities with an arbitrary number of moduli. We also discuss the notion of modality in the algebraic setting, its behaviour under morphisms, and its relations to formal deformation theory. As an application we show that the modality is semicontinuous in any characteristic.